Positive Borel measures

This is a note of real and complex analysis chapter 2.

Chapter 2 is about measures. The measure already defined in chapter 1. In chapter 2, every linear functionals, not combination, of a continuous function space on compact set (C) (Λf) represents the integration of the function (fdu) (Riesz representation theorem). Let X be a locally compact Hausdorf space, and let Λ be a positive linear functional on Cc(X). Then there exist a σalgebra in X which contains all Borel sets in X, and there exists a unique positive measure mu on M which represents Λ in the sense that (a) Λf=fdμ for every fCc(X) and following additional properties:
(b) μ(K)< for every compact set KX.
(c) For every EM, we have μ(E)=inf{μ(V):EinV,Vopen}.
(d) The relation μ(E)=sup{μ(K):KE,Kcompact}
holds for every open set E, and for every EM with μ(E)<.
(e) If EM,AsubsetE, and μ(E)=0, then AM.

The Riesz theorem is about linear functional Λ is equivalently replaced with choosing measure μ(E)=sup{Λf:fV}. Note sup{01f(x)dx=Λf:fV,V(0,1)}=1. The notion of include 0f1.

I confused Cc(X)

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Jun Kang
Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.

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